Page 79 - Modern Control Systems

Section 2.2 Differential Equations of Physical Systems 53
Nomenclature

3 Through-variable: F = force, T = torque, i = current, Q = fluid volumetric flow
rate, q = heat flow rate.
CI Across-variable: v = translational velocity, a> = angular velocity, v = voltage,
P = pressure, 9" = temperature.
• Inductive storage: L = inductance, \/k - reciprocal translational or rotational
stiffness, I = fluid inertance.
• Capacitive storage: C = capacitance, M - mass, J = moment of inertia, C/ = fluid
capacitance, C, = thermal capacitance.
3 Energy dissipators: R = resistance, b = viscous friction, R f = fluid resistance,
R, = thermal resistance.
The symbol v is used for both voltage in electrical circuits and velocity in trans-
lational mechanical systems and is distinguished within the context of each differen-
tial equation. For mechanical systems, one uses Newton's laws; for electrical systems,
Kirchhoff s voltage laws. For example, the simple spring-mass-damper mechanical
system shown in Figure 2.2(a) is described by Newton's second law of motion. (This
system could represent, for example, an automobile shock absorber.) The free-body
diagram of the mass M is shown in Figure 2.2(b). In this spring-mass-damper exam-
ple, we model the wall friction as a viscous damper, that is, the friction force is
linearly proportional to the velocity of the mass. In reality the friction force may be-
have in a more complicated fashion. For example, the wall friction may behave as a
Coulomb damper. Coulomb friction, also known as dry friction, is a nonlinear func-
tion of the mass velocity and possesses a discontinuity around zero velocity. For a
well-lubricated, sliding surface, the viscous friction is appropriate and will be used
here and in subsequent spring-mass-damper examples. Summing the forces acting
on M and utilizing Newton's second law yields

M-~- + b ^ - + ky(t) = r(t% (2.1)
dt* dt

where k is the spring constant of the ideal spring and b is the friction constant. Equa-
tion (2.1) is a second-order linear constant-coefficient differential equation.

Wall
5 _ i
friction, b
\ Mass
M
1
FIGURE 2.2
(a) Spring-mass- / • ( / )
damper system. Force
(b) Free-body
diagram. in)